A semiclassical, quantitative approach to the Anderson transition

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A semiclassical, quantitative approach to the
Anderson transition

Antonio M. García-García ag3_at_prin
ceton.edu Princeton University We study
analytically and numerically the metal-insulator
transition in a d dimensional, non interacting
(short range) disordered system by combining the
self-consistent theory of localization with the
one parameter scaling theory. 
The upper critical dimension is
infinity. Level Statistics at the transition

AGG, Emilio
Cuevas, Phys. Rev. B 75, 174203 (2007), AGG
arXiv0709.1292

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Main Goals
1. For a given disorder, E and d, how is the
quantum dynamics? Metal or insulator like? 2.
When does a metal insulator transition occurs?.
How is it described?
Insulator
Transition
Metal
Abs. Continuous
Pure point spectrum
Singular
Multifractal
Wigner-Dyson statistics
Poisson statistics
Critical statistics
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What we (physicists) know believe

• d 1 An insulator for any disorder
• d 2 An insulator for any disorder
• d gt 2 Disorder strong enough
  Insulator
• Disorder weak enough
  Mobility edge

r
Metal Insulator Transition
Why?
E
4
Approaches to localization
1. Numerical simulations
Currently reliable
Weak disorder/localization. Perturbation
theory. Well understood. Relevant in the
transition in d 2e (Wegner, Hikami, Efetov)
2. Analytical

• 1. Self-consistent theory from the insulator
  side, valid only for d gtgt2.
  No interference. Abou-Chacra, Anderson.
• 2. Self-consistent theory from the metallic
  side, valid only for d 2.
  No tunneling. Vollhardt
  and Wolfle.
• 3. One parameter scaling theory. Anderson et
  al.(1980) Correct
  (?) but qualitative.

Strong disorder/localization. NO quantitative
theory but
Some of the main results of the field are already
included in the original paper by Anderson 1957!!
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Energy Scales

• 1. Mean level spacing
• 2. Thouless energy
• tT(L) is the travel time to cross a box of size
  L

Dimensionless
Thouless conductance
Diffusive motion without
quantum corrections
Metal
Insulator
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Scaling theory of localization
The change in the conductance with the system
size only depends on the conductance itself

• 
  
  

g
Weak localization
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Predictions of the scaling theory at the
transition

1. Diffusion becomes anomalous
Imry, Slevin
2. Diffusion coefficient become size and momentum
dependent
3. ggc is scale invariant therefore level
statistics are scale invariant as well
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1.Cooperons (Langer-Neal, maximally crossed,
responsible for weak localization) and Diffusons
(no localization, semiclassical) can be
combined. 2. Perturbation theory around the
metallic limit. 3. No control on the
approximation.
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Perturbation theory around the insulator limit
(locator expansion).
No control on the approximation. Exact for a
Cayley tree. It should be a good approx for dgtgt2.
The distribution of the self energy Si (E) is
sensitive to localization.
metal
insulator
gt 0
metal
0
insulator
h
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Predictions of the self consistent theory
Correctly predicts a transition for dgt2
1. Critical exponents
Vollhardt, Wolfle
d 4 Upper critical dimension!
2. Critical disorder
Wc3d 14
Kroha, Wolfle, Kotov, Sadovskii
Anderson, Abou Chacra, Thouless
3. Critical conductance
also B. Shapiro and E. Abrahams 1980
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var
Numerical results at the transition
1. Scale invariance of the spectral
correlations.

2.
Intermediate level statistics 3. Critical
exponents 4. Critical disorder 5. Anomalous
diffusion





Mirlin, Evers, Cuevas, Schreiber, Slevin
Finite scale analysis, Shapiro, et al. 93
Agreement scaling theory
Insulator Metal
Schreiber, Grussbach
Disagreement with the selfconsistent theory !
?
Agreement scaling theory
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SECOND PART
What we did
1. Numerical results for the Anderson transition
in d4,5,6, AGG and E. Cuevas, Phys. Rev. B 75,
174203 (2007), Critical exponents, critical
disorder, level statistics
2. Analytical results combining the scaling
theory and the self consistent condition, AGG,
arXiv0709.1292 Critical exponents, critical
disorder, level statistics.
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Numerical Results Anderson model cubic lattice,
d3,6
Metal Insulator

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Critical exponents and Critical Disorder
Wc/ln(Wc/2)

• 
  
• 
  Cayley tree
• Upper critical dimension is infinity

OK but
Self consistent theory
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Level Statistics
ln(P(s))
A(d)-1
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Analytical results
Why do self consistent methods fail for d ? 3?

• 1. Always perturbative around the metallic
  (Wolhardt Wolfle) or the insulator state
  (Anderson, Abou Chacra, Thouless) .
• A new basis for localization is needed
• 2. Anomalous diffusion at the transition
  (predicted by the scaling theory) is not taken
  into account.

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Idea! (AGG arXiv0709.1292) Solve the self
consistent equation assuming that the diffusion
coefficient is renormalized as predicted by the
scaling theory
Assumptions
1. All the quantum corrections missing in the
self consistent treatment are included by just
renormalizing the coefficient of diffusion
following the scaling theory.

• 2. Right at the transition the quantum dynamics
  is well described by a process of anomalous
  diffusion. with no further localization
  corrections.

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Technical details Critical exponents
2
The critical exponent ?, can be obtained by
solving the above equation for
with D (?) 0.
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Level Statistics
Starting point Anomalous
diffusion predicted by the scaling theory
Semiclassically, only diffusons
Two levels correlation function
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Cayley tree
Aizenman, Warzel
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Comparison with numerical results
1. Critical exponents Excellent 2, Level
statistics Good (problem with gc) 3. Critical
disorder Not better than before
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CONCLUSIONS

• 1.We obtain analytical results at the transition
  by combining the scaling theory with the self
  consistent in dgt3.
• 2. The upper critical dimension is infinity
• 3. Analytical results on the level statistics
  agree with numerical simulations.

What is next?
1. Experimental verification. 2. Anderson
transition in correlated potential
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Experiments

• Our findings may be used to test experimentally
  the Anderson transition by using ultracold atoms
  techniques.
• One places a dilute sample of ultracold Na/Cs in
  a periodic step-like standing wave which is
  pulsed in time to approximate a delta function
  then the atom momentum distribution is measured.
• The classical singularity cannot be reproduced
  in the lab. However (AGG, W Jiao, PRA 2006) an
  approximate singularity will still show typical
  features of a metal insulator transition.
  

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• Spectral signatures of a metal (Wigner-Dyson)
  
• 1. Level Repulsion
• 2. Spectral Rigidity
• 
  
• Spectral Signatures of an insulator (Poisson)
• 
  

P(s)
s